> How do you keep all your knowledge from your mathematics courses? So> many different topics are presented, learned, then mostly forgotten as> you study in a different course. Overlap of topics is not so great.

I once thought that yoiu could make a diagram of "math." It would look something like a family tree, with branches. Counting would be the root, followed by addition and subtraction, and so on. As you got farther in math, the topics, it seemed, spread out all over the place so this seemed like the best model. Without filling in details, something like:

| | ^ / \ ^ \ / \ ^ ^ | / \> Do some of the middle-division level courses> include a review of stuff that was previously studied in the lower> division course?

As I continued on in my studies, right now working towards a Ph.D., I have come to realize that the above diagram is way too simple. It seems farther you get in math, the more realize the topics do overlap. For example, group theory is very important in combinatorics. Certain topics in linear algebra have analogues in analysis and topology. Math modeling, one of the most applied aspects of math, relies heavily on partial differential equations, which relies on analysis, one of the most "pure" parts of math.

There might not be a formal review in mid- and upper-level math classes, but you are generally expected to remember concepts and be able to "remember" everything, even if you have to look up one or two things. That is, you probably don't need to remember all 15 or so ways to tell in a matrix is invertible or not in most classes outside of introductory linear algebra. But if it came up in a class, like group theory, you should be able to know where to find the information and be able to remember some of the most common, most useful ones (like non-zero determinant).

> People who earn a university degree in Mathematics must be able to> hold together many different skills and concepts and not forget them> if these people expect to enter a Master's program. What is the> reality? How do people hold together all those skills and topics?

The nice thing about a Master's program, and any post-bachelor degree program is that you spend almost all of your time doing math. No need for silly things like history or literature[*] to take away from your time with math. The second nice thing is that usually, usually you concentrate on one or two specific areas of math to study.

Right now I can't tell you the defining properties of a sigma-algebra, even though I have taken courses where it was required for me to know them. I do have texts I can look this information up in. However, I can, however, tell you properties that will make Ulam's Conjecture true. It turns out one of my areas of concentration is graph theory and combinatorics.